A parameterized curve is a mathematical representation of a curve in which the coordinates of points on the curve are expressed as functions of one or more independent parameters, most commonly a single real parameter, usually denoted by .

Finding Parameterizations of Common Curves

Lines

Let be the line passing through the point that is parallel to the vector . We can then parameterize as follows:

Line Segments

Let be the line from the points to . We can the parameterize as follows:

Circles

Given a circle of radius centred at the origin in the -plane. We can parameterize the circle as follows:

Ellipse

Given a ellipse with two points at and centred at the origin in the -plane. We can parameterize the ellipse as follows:

Examples

Example 1

Find the parameterization of the curve given by the intersection of the cylinder and the plane .

The projection of onto the -plane is the unit circle so we can take

Then since we get

Example 2

Find a parameterization of the curve given by the intersection of the surfaces and .

TODO